Instituto de Física da Universidade de São Paulo, São Carlos
“Projeto Café com Física”
Relação entre elasticidade de DNA e a ligação cooperativa de proteínas e
fármacos
Oscar Nassif Mesquita
Departamento de Física, Universidade Federal de Minas Gerais, Belo Horizonte
Trabalho em colaboração com:
Lívia Siman Gomes (Doutoranda, Física - UFMG)
Ismael S. Silva Carrasco (Mestrando, Física - UFV)
Prof. Jafferson K. L. da Silva (Física - UFMG)
Prof. Ricardo S. Schor (Física – UFMG)
Profa. Mônica C. de Oliveira (Farmácia – UFMG)
Prof. Márcio Santos Rocha (Física – UFV)
Agências financiadoras:
Fapemig, CNPq, Pronex-Facepe, INCFx-Instituto Nacional de Fluidos Complexos e
Aplicações
Outline
•
Stretching single DNA molecules with optical tweezers: measurement of the
persistence length and contour length.
•
Study of the interaction between DNA and molecules of pharmaceutical interest.
•
Interaction between DNA and beta-cyclodextrin: non-monotonic flexibility.
•
HU-DNA interaction: previous example of non-monotonic flexibility.
•
Hill cooperativity in biochemical reactions.
•
Our two-sites quenched disorder model to explain non-monotonic flexibilities.
•
Results and discussion.
•
Conclusions.
An optical tweezers is just a light beam trapping some material
(A. Ashkin example)
Single Molecule Experiments
Schematic set-up of optical tweezers
Optical tweezers is an invention of A. Ashkin in 1970, Phys. Rev. Lett. 24, 156 (1970)
Complete theory of optical tweezers for
dielectric spheres by Maia Neto and
Nussenzveig (Europhys. Lett, 50, 70C2
(2000)), and Mazolli, Maia Neto and
Nussenzveig (Proc. R. Soc. Lond. A 459,
3021 (2003)), named Mie-Debye (MD)
theory.
Viana, Rocha, Mesquita, Mazolli, Maia
Neto, and Nussenzveig, APL (2006), and
PRE (2007).
Set – up at
UFMG
Brownian motion of a microsphere in a
harmonic potential
Langevin equation:
d 2 x dx
m 2
 kx  f (t )
dt
dt
 f (t ) f (t )  2k BT (t  t )
Position correlation function satisfies the Langevin equation:
d 2  x(0) x(t ) 
d  x(0) x(t ) 
m

 k  x(0) x(t )  0
2
dt
dt
Neglecting inertia and using the equipartition theorem
k BT
 x 
ki
2
xi 0 xi t  

k BT
e
ki
6a
i  
ki
ki

t
i
Time autocorrelation function of
back-scattered intensity
fluctuations of a trapped bead
Intensity back-scattering profile
From the time autocorrelation function we obtain the tweezers´
stiffness for motion perpendicular and parallel to the incident
pN
direction.
k  5 .8  0 .2
x
m
Tweezers calibration with video-imaging
k = 0.4 pN/m
X
Normalized histogram
5
20
4
3
Potential energy
Probability
15
10
5
2
1
0
-1
0
-2
-3
0,5
-5
0,5
0,55
0,6
0,65
0,7
0,75
0,8
0,55
0,6
0,65
0,7
0,85
X (m)
X (m)
𝐸
𝑘𝐵 𝑇
=-ln (probability)
0,75
0,8
0,85
Entropic elasticity of a single DNA molecule
DNA and RNA stretching experiments
First experiment by Carlos Bustamante and co-workers Science (1992)
Nucleotides
Adenine, Guanine, Cytosine, Tymine
Stretching DNA : entropic elasticity
𝑡 0 . 𝑡(𝑠) =
𝑠
−
𝑒 𝐴
A is the polymer persistence length
A = bending rigidity/thermal energy




k T z
1
1
F B  

2
A L
4
z

41  


 L


Marko and Siggia expression for
the entropic force, where A is the
persistence length, z is the end-toend distance and L is the contour
length of the polymer.
A  50  1 nm
L  19.5  0.1 m
Viana, Freire & Mesquita, PRE 65, 041921 (2002)
DNA/Ethidium Bromide
Fit to the neighbor exclusion model
DNA-psoralen interaction
Persistence length with and without
UV light
Relative increase of contour length
ab initio DFT calculations
Psoralen-DNA fragment with five base CG pairs and two
intercalated psoralens obtained from our ab initio DFT calculations.
DNA-psoralen: Single-molecule experiments and first principles calculations, APL (2009)
M. S. Rocha, A. D. Lúcio, S. S. Alexandre, R. W. Nunes, and O. N. Mesquita
Cyclodextrins are used for condensing DNA
into small lipid vesicles for gene therapy
CD-DNA persistence length measured with optical tweezers
Blue squares – cationic CD
Red circles – neutral CD
HU-DNA persistence length measured with magnetic tweezers
(continuous curve is a guide to the eye)
total HU concentration (nM)
J. van Noort et al., PNAS 101 (18), 6969 (2004)
HU-DNA model for binding and DNA structural changes
Sagi et al., J. Mol. Biol., 341, 419 (2004)
smaller persistence length
larger persistence length
A) HU dimmers (spheres) bind cooperatively (bound-clusters with 4 or 5 HU molecules as
measured by FRET) and compacts DNA at low protein concentration, each HU dimmer
introducing a small local bend.
B) At high HU concentrations, compactation by HU is reversed, and the protein appears to form
a complex with helical structure with DNA.
A mechanism of interaction of CD and DNA with a flipping-out DNA base
M. A. Spies and R. L. Schowen, J. Am. Chem. Soc. 124, 14049 (2002)
Hill cooperativity
n ligands bind simultaneously to the substrate (bound-cluster)
𝑛𝐿 + 𝑆
L for ligand and S for substrate
Mass-action law: K =
𝐿𝑛 𝑆
𝑆 𝐿𝑛
Fraction of ligands bound: 𝑟
=𝐶
𝑟
𝑚𝑎𝑥
𝐶𝑏
𝑛
𝑏𝑝 𝐶𝑓
=
𝑟
, 𝑟
𝑚𝑎𝑥
=𝐶
𝐾𝐶𝑏𝑝 𝐶𝑓 𝑛
𝑏𝑝 +𝐾𝐶𝑏𝑝 𝐶𝑓
𝑛
𝐾𝐶𝑓 𝑛
= 1+𝐾𝐶
𝑓
𝐶𝑓 𝑛
𝑛
𝐾𝑑
=
1+
𝐶𝑓 𝑛
𝐾𝑑
Hill binding isotherm
1
1
=2
Hill exponent
n < 1 negative cooperativity
n = 1 non-cooperativity
n > 1 positive cooperativity
0,8
fraction bound
for 𝐶𝑓 = 𝐾𝑑
𝐿𝑛 𝑆
(chemical constant)
𝐶𝑏
𝐶𝑏𝑝 +𝐶𝑏
𝐾𝑑 is the dissociation constant;
𝐾
0,6
n=1
n=4
n=10
0,4
0,2
𝐾𝑑 = 40
0
0
20
40
60
C
f
80
100
Two-sites quenched disorder model
1) Assumption 1: When a bound-cluster binds to DNA it decreases the persistence length from
the bare DNA value 𝐴0 to 𝐴1 ; if two bound-clusters become nearest-neighbors they stiffen
the DNA, resulting in a larger persistence length 𝐴2 .
2) Assumption 2: The bound-clusters have the same average size of n molecules, cannot move
along the DNA (quenched disorder), and are randomly distributed along the DNA. As one
increases the ligand concentration in solution, the number of clusters increases
proportionally, but not their size.
a) Two sites empty, 𝐴0 , have probability 𝑃0 = 1 − 𝑟 𝑟𝑚𝑎𝑥 2 ;
b) One site empty and the other occupied, 𝐴1 , have probability 𝑃1 = 2𝑟 𝑟𝑚𝑎𝑥 1 − 𝑟 𝑟𝑚𝑎𝑥 ;
c) Two sites occupied, 𝐴2 , have probability 𝑃2 = 𝑟 𝑟𝑚𝑎𝑥 2 .
Resulting equation for the model
1 𝑃0 𝑃1 𝑃2
1
2
2
𝑟
1
2
1
=
+
+
=
+
−
+
−
+
𝐴 𝐴0 𝐴1 𝐴2 𝐴0
𝐴1 𝐴0 𝑟𝑚𝑎𝑥
𝐴0 𝐴1 𝐴2
𝑟
𝑟𝑚𝑎𝑥
𝐶𝑓 𝑛
𝐾𝑑
=
𝐶𝑓
1+ 𝐾
𝑑
𝑛
≡ 𝐻 𝐶𝑓
𝑟
2
𝑟𝑚𝑎𝑥
𝐴0 = 50𝑛𝑚
𝐴1 , 𝐴2 , 𝐾𝑑 , 𝑛 free adjustable parameters
Solving Hill equation iteratively
Equation has a single-fixed point solution
Experimentally we know the total ligand concentration 𝐶𝑇 but not the free ligand concentration
𝐶𝑓 . Since 𝐶𝑓 = 𝐶𝑇 − 𝐶𝑏 = 𝐶𝑇 − 𝑟𝐶𝑏𝑝 then,
𝑎 = 1 𝑏 = 0.5
1
0,8
𝐻 𝐶𝑓 = 𝐻 𝐶𝑇 − 𝑟𝐶𝑏𝑝 = 𝐻 𝐶𝑇 − 𝑟𝑚𝑎𝑥 𝐶𝑏𝑝 𝐻 𝐶𝑓
0,6
0,4
a) zeroth-order solution: 𝐻 𝐶𝑓 ≅ 𝐻 𝐶𝑇
0,2
b) first-order solution: 𝐻 𝐶𝑓 ≅ 𝐻 𝐶𝑇 − 𝑟𝑚𝑎𝑥 𝐶𝑏𝑝 𝐻 𝐶𝑇
0
0
0,2
0,4
0,6
0,8
1
x
B
𝑥 ≡ 𝑟 𝑟𝑚𝑎𝑥
𝑎 ≡ 𝐶𝑇 𝐾𝑑
𝑏 ≡ 𝑟𝑚𝑎𝑥 𝐶𝑏𝑝 𝐾𝑑
convergence criterium
𝑎 − 𝑏𝑥 𝑛
𝑥=
1 + 𝑎 − 𝑏𝑥
1.8
𝑛
=𝑓 𝑥
1.6
1.4
1.2
Iterative solution possible if
𝑑𝑓 𝑥
𝑑𝑥
<1
then
𝑏<
1
1
𝑛−1 𝑛
4𝑛 𝑛+1
𝑛2 −1
0.8
, ∀𝑎
0.6
0
1
2
3
4
n
5
6
7
Cationic CD-DNA interaction
Fit using our model with first-order Hill equation
0.08
0.06
𝐴1 = 7.9 ± 0.2 𝑛𝑚
0.05
𝐴2 = 125 ± 20 𝑛𝑚
0.04
𝐾𝑑 = 9.6 ± 0.3 𝜇𝑀
0.03
𝑛 = 3.45 ± 0.15
-1
-1
A (nm )
0.07
0.02
0.01
0
0
10
20
30
C (M)
T
40
50
60
HU-DNA interaction
Fit using our model with a zeroth-ordem Hill equation
Data from J. van Noort et al., PNAS 101 (18), 6969 (2004)
0.07
𝐴1 = 8.2 ± 0.2 𝑛𝑚
-1
0.04
𝐴2 = 125 ± 15 𝑛𝑚
0.03
𝐾𝑑 = 36.9 ± 0.9 𝑛𝑀
0.02
𝑛 = 3.2 ± 0.1
A (nm )
0.05
-1
0.06
0.01
0
1
10
100
C (nM)
T
1000
Conclusions
•
We can study DNA interactions with ligands by measuring the persistence length and
contour length of the complexes formed, using optical tweezers in single-molecule
assays.
•
Interaction between DNA and beta-cyclodextrin and between HU-DNA cause nonmonotonic persistence length behavior, indicating that for low ligand concentration
the complex formed is more flexible and for higher concentrations more rigid.
•
We propose a two-sites quenched disorder statistical model together with Hill
cooperativity, which provides a model function which fits very well both sets of data.
Our model predicts that the binding kinetics is mediated by size stabilized boundclusters. With the quantitative parameters obtained we were able to propose a
microscopic physical mechanism for the CD-DNA cooperative binding.
•
Therefore, from a single mechanical measurement we can obtain the elastic
parameters related to structural changes of the DNA molecule caused by the ligands,
together with the chemical parameters of the binding reaction.